Earthquakes occur from the creation and propagation of a rupture. Rapid motion produces seismic waves, which are usually observed in the far-field. Source inversion can use such observations to describe the kinematic behaviour of the source by the moment tensor, while accounting for measurement uncertainties.
For regional and global earthquakes, it is often feasible to pre-calculate databases of Green functions to perform full waveform inversions efficiently (Heimann 2011; Duputel et al. 2012b), although this can also be done on the fly for less complex velocity structures (e.g. Dziewonski et al. 1981; Ekström et al. 2012) and even for microseismic cases (e.g. O'Toole 2013). However microseismic earthquakes are small-magnitude earthquakes often detected on a small local network of receivers. For these cases, it is difficult to compute the Green functions database because the velocity structure is rarely well constrained and the event locations are often distributed throughout the network. Furthermore, velocity variations in the region may have a large effect on the ray paths due to the close proximity of the receivers to the sources. Therefore, observations such as P- and S-wave polarities and amplitude ratios are more robust in these regimes, so are commonly used to constrain the source inversion (Reasenberg & Oppenheimer 1985; Hardebeck & Shearer 2002, 2003; Snoke 2003). Such approaches still depend on knowing the velocity model to calculate the azimuths and take-off angles of the rays from the source to the instrument. Many of the current inversions provide a result and an estimate of some misfit or quality parameter.
There are several Bayesian approaches to source inversion. Often these are based on full waveform inversion approaches (e.g. Kennet et al. 2000; Wéber 2006), and can extend the approach to near-field observations and finite-fault models (e.g. Zollo & Bernard 2007; Minson et al. 2013), or other data sources (e.g. O'Toole 2013; Kaeufl et al. 2013).
The source function of an earthquake is a function of both time and position, although, in the case of microseismic events, the time dependence is generally modelled as a step function, so it is usually assumed that there is no spatial dependence of the time component. Therefore, the source function can be split into the time-dependent source-time function, S(t), and the spatially dependent moment tensor, M. The moment tensor describes the nine force couples required to define a point source (FIG. 1). While angular momentum is conserved, the moment tensor must be symmetrical and, therefore, there are six independent components.
In most cases, especially for teleseismic events, the moment tensor appears to be double-couple or close to double-couple, signifying slip along a fault plane. Some non-double-couple mechanisms have been observed, such as those associated with nuclear explosions (Müller 1973; Ford et al. 2008), and potentially non-double-couple mechanisms in events in volcanic and geothermal regions and other areas associated with induced seismicity, such as hydraulic fracturing (Vasco 1990; Foulger et al. 2004; Templeton & Dreger 2006; Vavryčuk et al. 2008). These non-double-couple mechanisms could arise from processes such as conduit collapse or fracture opening, perhaps associated with fluid movement. However, apparent (but incorrect) non-double-couple characteristics can also be caused by uncertainties in the inversion such as noise in the data or velocity model uncertainties, finite fault effects (Kuge & Lay 1994), as well as the improved fit due to the extra two parameters in the full moment tensor compared to the double-couple source (Panza & Sarao 2000).
The six independent moment tensor components can be written as a six-vector, in a form similar to the Voigt form (Voigt 1910) and that of Chapman & Leaney (2011). Using this notation, it is possible to write the far-field seismic amplitude equation as shown in eq. (1), where a is the matrix of station propagation coefficients, aSTAi (a derivation of the isotropic propagation coefficients can be found in Appendix A), u is the vector of amplitudes recorded at the receiver stations STAi and {tilde over (M)} is the six-vector form of the moment tensor M.
                    u        =                              (                                                                                u                                          STA                      1                                                                                                                                        u                                          STA                      2                                                                                                                                        u                                          STA                      3                                                                                                                                        u                                          STA                      4                                                                                                                                        u                                          STA                      5                                                                                                                    ⋮                                                      )                    =                                                    (                                                                                                    a                                                  STA                          1                                                                                                                                                                        a                                                  STA                          2                                                                                                                                                                        a                                                  STA                          3                                                                                                                                                                        a                                                  STA                          4                                                                                                                                                                        a                                                  STA                          5                                                                                                                                                ⋮                                                                      )                            ⁢                              (                                                                                                    M                        11                                                                                                                                                M                        22                                                                                                                                                M                        33                                                                                                                                                                          2                                                ⁢                                                  M                          23                                                                                                                                                                                                  2                                                ⁢                                                  M                          13                                                                                                                                                                                                  2                                                ⁢                                                  M                          12                                                                                                                    )                                      =                          a              ·                                                M                  ~                                .                                                                        (        1        )            
The six-vector components are scaled in eq. (1) so that the six-vector can be normalized to give a normalized moment tensor (eq. 2), defined following Chapman & Leaney (2011):
                                          ∑                          i              ,                              j                =                1                                      3                    ⁢                                          ⁢                      M            ij            2                          =        1.                            (        2        )            
Because eq. (1) is linear, a suitable left pseudo-inverse can be calculated and the moment tensor determined. The seismic amplitude equation (eq. 1) can also represent the full waveform problem, with u signifying the vector of waveforms and the station propagation coefficients (a), now time dependent and containing the Green functions and station responses. This is again a linear problem for known Green functions.
When using first-motion polarity information (Y) for source inversion, the signum function prevents a linear inversion approach:Y=sgn(a·{tilde over (M)}).  (3)
For amplitude inversions, the propagation effects are dependent on the velocity structure and attenuation models used, whereas amplitude ratios are less dependent on these, and so are often used in preference to absolute amplitudes (Hardebeck & Shearer 2002, 2003; Snoke 2003). While it is possible to invert directly for the moment tensor using amplitudes or amplitude ratios, multiple different observation types can make it difficult to invert for the source directly, and impossible when using the easily measured first-motion polarities due to the non-linear signum function (eq. 3).